This text deals with A¹-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundational paper on A¹-homotopy theory written together with V. Voevodsky. Inspired by classical results in algebraic topology, we present new techniques, new results and applications related to the properties and computations of A¹-homotopy sheaves, A¹-homology sheaves, and sheaves with generalized transfers, as well as to algebraic vector bundles over affine smooth varieties.

1 Introduction.- 2 Unramified sheaves and strongly A¹-invariant sheaves.- 3 Unramified Milnor-Witt K-theories.- 4 Geometric versus canonical transfers.- 5 The Rost-Schmid complex of a strongly A¹-invariant sheaf.- 6 A¹-homotopy sheaves and A¹-homology sheaves.- 7 A¹-coverings.- 8 A¹-homotopy and algebraic vector bundles.- 9 The affine B.G. property for the linear groups and the Grassmanian